Categories and Functors

280 Pages·1970·3.47 MB·English

by  Bodo Pareigis

CATEGORIES
AND FUNCTORS
This is Volume 39 in
PURE AND APPLIED MATHEMATICS
A Series of Monographs and Textbooks
Editors: PAULA . SMITH AND SAMUEL EILENBERG
A complete list of titles in this series appears at the end of this volume
CATEGORIES
AND FUNCTORS
Bodo Pareigis
UXIVERSITY OF MUNICH
MUNICH, GERMANY
1970
ACADEMIC PRESS New York London
This is the only authorized English translation of Kuregorien
-
und Funktoren Eine Einfirhrung (a volume in the series
“Mathematische Leitfaden,” edited by Professor G. Kothe), orig-
inally in German by Verlag B. G. Teubner, Stuttgart. 1969
COPYRIGHT 0 1970, BY ACADEMPIRCE SS, INC.
ALL RIGHTS RESERVED
NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM,
BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY
OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM
THE PUBLISHERS.
ACADEMIC PRESS, INC.
111 Fifth Avenue, New York, New York 10003
United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
Berkeley Square House, London WlX 6BA
-
LIBRAROYF CONGRESS CATALOG CARD NUMBER76: 117631
PRINTED IN THE UNITED STATES OF AMERICA
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
.
1 Preliminary Notions
1.1 Definition of a Category . . . . . . . . . . . . . . . . . 1
. . . . . . . . . .
1.2 Functors and Natural Transformations 6
1.3 Representable Functors . . . . . . . . . . . . . . . . . 10
1.4 Duality . . . . . . . . . . . . . . . . . . . . . . . 12
. . . .
1.5 Monomorphisms. Epimorphisms. and Isomorphisms 14
. . . . . . . . . . . . .
1.6 Subobjects and Quotient Objects 20
. . . . . . . . . . . .
1.7 Zero Objects and Zero Morphisms 22
1.8 Diagrams . . . . . . . . . . . . . . . . . . . . . . . 24
1.9 Difference Kernels and Difference Cokernels . . . . . . . . 26
. . . . . . . . . . . . . . . .
1.10 Sections and Retractions 29
1.11 Products and Coproducts . . . . . . . . . . . . . . . . 29
. . . . . . . . . . . . . . . .
1.12 Intersections and Unions 33
. . . . . . . . . .
1.13 Images. Coimages. and Counterimages 34
1.14 Multifunctors . . . . . . . . . . . . . . . . . . . . . 39
1.15 The Yoneda Lemma . . . . . . . . . . . . . . . . . . 41
1.16 Categories as Classes . . . . . . . . . . . . . . . . . . 48
Problems . . . . . . . . . . . . . . . . . . . . . . . 49
.
2 Adjoint Functors and Limits
2.1 Adjoint Functors . . . . . . . . . . . . . . . . . . . . 51
2.2 Universal Problems . . . . . . . . . . . . . . . . . . . 56
2.3 Monads . . . . . . . . . . . . . . . . . . . . . . . 61
2.4 Reflexive Subcategories . . . . . . . . . . . . . . . . . 73
. . . . . . . . . . . . . . . . . .
2.5 Limits and Colimits 77
2.6 Special Limits and Colimits . . . . . . . . . . . . . . . 81
2.7 Diagram Categories . . . . . . . . . . . . . . . . . . 89
V
vi CONTENTS
2.8 Constructions with Limits . . . . . . . . . . . . . . . . 97
2.9 The Adjoint Functor Theorem . . . . . . . . . . . . . . 105
2.10 Generators and Cogenerators . . . . . . . . . . . . . . 110
2.11 Special Casesof the Adjoint Functor Theorem . . . . . . . 113
2.12 Full and Faithful Functors . . . . . . . . . . . . . . . 115
Problems . . . . . . . . . . . . . . . . . . . . . . . 118
.
3 Universal Algebra
3.1 Algebraic Theories . . . . . . . . . . . . . . . . . . 120
3.2 Algebraic Categories . . . . . . . . . . . . . . . . . . 126
3.3 Free Algebras . . . . . . . . . . . . . . . . . . . . . 130
3.4 Algebraic Functors . . . . . . . . . . . . . . . . . . . 137
. . . . . . .
3.5 Examples of Algebraic Theories and Functors 145
3.6 Algebras in Arbitrary Categories . . . . . . . . . . . . . 149
. . . . . . . . . . . . . . . . . . . . . . .
Problems 156
.
4 Abelian Categories
4.1 Additive Categories . . . . . . . . . . . . . . . . . . 158
4.2 Abelian Categories . . . . . . . . . . . . . . . . . . . 163
4.3 Exact Sequences . . . . . . . . . . . . . . . . . . . . 166
4.4 Isomorphism Theorems . . . . . . . . . . . . . . . . . 172
4.5 The Jordan-Holder Theorem . . . . . . . . . . . . . . 174
4.6 Additive Functors . . . . . . . . . . . . . . . . . . . 178
4.7 Grothendieck Categories . . . . . . . . . . . . . . . . 181
4.8 The Krull.Remak.Schmidt.AzumayaTheorem . . . . . . . 190
4.9 Injective and Projective Objects and Hulls . . . . . . . . . 195
4.10 Finitely Generated Objects . . . . . . . . . . . . . . . 204
4.1 1 Module Categories . . . . . . . . . . . . . . . . . . . 210
. . . . . . . . . . . . . .
4.12 Semisimple and Simple Rings 217
4.13 Functor Categories . . . . . . . . . . . . . . . . . . . 221
. . . . . . . . . . . . . . . . .
4.14 Embedding Theorems 236
Problems . . . . . . . . . . . . . . . . . . . . . . . 244
Appendix . Fundamentals of Set Theory . . . . . . . . . . 2 47
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . 257
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
Thinking-is it a social function
or one of the brains 7
Stanislaw Jmzy Lec
Preface
In their paper on a “General theory of natural equivalences” Eilenberg
and MacLane laid the foundation of the theory of categories and
functors in 1945. It took about ten years before the time was ripe for
a further development of this theory. Early in this century studies of
isolated mathematical objects were predominant. During the last
decades, however, interest proceeded gradually to the analysis of
admissible maps between mathematical objects and to whole classes of
objects. This new point of view is appropriately expressed by the
theory of categories and functors. Its new language-originally called
“general abstract nonsense” even by its initiators-spread into many
different branches of mathematics.
The theory of categories and functors abstracts the concepts “object”
and “map” from the underlying mathematical fields, for example, from
algebra or topology, to investigate which statements can be proved in
such an abstract structure. Then these statements will be true in all
those mathematical fields which may be expressed by means of this
language.
Of course, there are trends today to render the theory of categories
and functors independent of other mathematical branches, which will
certainly be fascinating if seen for example, in connection with the
foundation of mathematics. At the moment, however, the prevailing
value of this theory lies in the fact that many different mathematical
fields may be interpreted as categories and that the techniques and
theorems of this theory may be applied to these fields. It provides the
means of comprehension of larger parts of mathematics. It often occurs
that certain proofs, for example, in algebra or in topology, use “similar”
methods. With this new language it is possible to express these “simi-
larities” in exact terms. Parallel to this fact there is a unification. Thus
it will be easier for the mathematician who has command of this language
to acquaint himself with the fundamentals of a new mathematical field
if the fundamentals are given in a categorical language.
vii
viii PREFACE
This book is meant to be an introduction to the theory of categories
and functors for the mathematician who is not yet familiar with it,
as well as for the beginning graduate student who knows some first
examples for an application of this theory. For this reason the first
chapter has been written in great detail. The most important terms
occurring in most mathematical branches in one way or another have
been expressed in the language of categories. The reader should consider
the examples-most of them from algebra or topology-as applications
as well as a possible way to acquaint himself with this particular field.
The second chapter mainly deals with adjoint functors and limits
in a way first introduced by Kan.
The third chapter shows how far universal algebra can be represented
by categorical means. For this purpose we use the methods of monads
(triples) and also of algebraic theories. Here you will find represented
one of today’s most interesting application of category theory.
The fourth chapter is devoted to abelian categories, a very important
generalization of the categories of modules. Here many interesting
theorems about modules are proved in this general frame. The em-
bedding theorems at the end of this chapter make it possible to transfer
many more results from module categories to arbitrary abelian categories.
The appendix on set theory offers an axiomatic foundation for the
set theoretic notions used in the definition of categorical notions. We
use the set of axioms of Godel and Bernays. Furthermore, we give a
formulation of the axiom of choice that is particularly suitable for an
application to the theory of categories and functors.
I hope that this book will serve well as an introduction and, moreover,
enable the reader to proceed to the study of the original literature. He
will find some important publications listed at the end of this book,
which again include references to the original literature.
Particular thanks are due to my wife Karin. Without her help in
preparing the translation I would not have been able to present to
English speaking readers the English version of this book.